new Divifion of ibe Quadrant. 31 
and thus a number of primary fines, &c. may be found, to 
check or verify the fame when computed by other methods. By 
this method will be found the fines, &c. of the arcs 
I, 10, 100, 1000, 19000, 100000; 
2, 20, 200, 2000, 20000; 
35 30, 300, 3000, 30000 ; 
4, 40, 400, 4000, 40000 ; 
é&c. till 
99 992, 9900, 99000, 990000. © 
13. Again, it is evident, that, of the terms in the feries for 
the fine, the firft term ¢ alone will give the fine true to the 
neareft unit in the ninth place in the firft 144 fines, or the arc 
and fine will be the fame for nine places as. far as the arc 1443 
but they will agree to the neareft unit in the feventh place as 
far as the arc 669; after which the fecond term of the feries 
mutt be included. 
‘ 34. When the fecond term is taken in, thefe two terms 
a —+ a will give the fines true tothe neareft unit in the ninth 
place till the arc becomes 3500. Now the numbers in my 
table of cubes (juit publifhed by order of the Board of Longi- 
tude) extend to 10000, and therefore all the above cubes are 
found in it; confequently taking the fixth part of thofe cubes, 
and fubtracting it from the correfponding arcs, the remainders. 
will be the fines of thofe arcs, as far as till the are be 3500: 
after which the third term of the feries may be taken in, or 
other methods may be ufed. 
15. But fince, for any arc a, this is a general theorem, wz. 
as radius : 2 cof. a:: fin. za: fin.a—1xa+fin.u+1xa; tak- 
ing @=1, radius 10000, the fine of 2 will be 1 — oog00000004 
and the cofine of @ will be 1000e0-:‘o00005, and the 
eel above 
